Some people dislike the fact that they get older every year, which makes them not want to celebrate their birthday.
Introverts don't like big gatherings and lots of attention, which makes them dislike celebrating their birthday.
Some people don't like being the center of attention, and since birthdays are all about attention, they might feel uncomfortable celebrating it.
Antisocial people prefer to be alone most of the time. They might celebrate their birthday by going on a walk alone, for example, but they would not throw a birthday party.
Some people think big gatherings are impersonal since some people might just be there to have fun regardless of whose birthday it is.
People who think age doesn't really matter and that it's just a number often don't celebrate their birthday.
Some people don't understand why growing older is an occasion to be celebrated. See Why do people celebrate their birthdays
People who are not materialistic don't care about birthday presents and might even prefer not receiving any gifts at all.
Birthday parties are all about inviting many people and loud music, which some people don't like since not everyone likes big social gatherings.
Some people may not like celebrating their birthday if it reminds them of a certain bad event that happened on the same day such as the death of someone close.
Birthdays nowadays have become a show of fake sentiments due to social media networks. People on Facebook often receive birthday wishes from people they no longer talk to, for example, which some might find as an insincere gesture.